LINEAR RATIONAL INSURANCE MODEL & ECONOMIC SCENARIO GENERATOR

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SCUOLA DI DOTTORATO

UNIVERSITÀ DEGLI STUDI DI MILANO-BICOCCA

Dipartimento di / Department of

STATISTICA E METODI QUANTITATIVI

Dottorato di Ricerca in / PhD program STATISTICA E FINANZA MATEMATICA

Ciclo / Cycle XXXII

Curriculum in

MATEMATICA PER LA FINANZA

LINEAR RATIONAL INSURANCE MODEL

&

ECONOMIC SCENARIO GENERATOR

Cognome / Surname

DEL GUSTO Nome / Name LUIGI CARMINE

Matricola / Registration number

746006

Tutore / Tutor: EMANUELA ROSAZZA GIANIN

Coordinatore / Coordinator: GIORGIO VITTADINI

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Ph.D. thesis

Subject: LINEAR RATIONAL INSURANCE MODEL

ECONOMIC SCENARIO GENERATOR

Del Gusto

Luigi Carmine

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This result is very important because it allows us, under the hypothesis that the diﬀusion part is a martingale, to use all the results we have about linear drift diﬀusions for this new set of variables. As a consequence, we are able to give the price of three important life insurance contracts: the sur-vival and death benefit and the guaranteed annuity option (also called GAO).

It is about the GAO that we can see the advantage of the framework we are using: actually the payoff of the GAO is not an affine or a polynomial function, so the only way to treat it is by performing a change of measure or a Monte Carlo simulation. We show that, under the assumption that the state space is compact, we are able to approximate the GAO payoff by a polynomial, which will allow us to find a closed formula for the price of this contract.

The end of this work is dedicated to some numerical experiments which have the aim to point out the importance of the choice of the degree of the approximated polynomials in order to have reliable results. We show that a ten degree polynomial is able to estimate with a small error the Monte Carlo price of the GAO.

This work extend the existing literature concerning polynomial models and their application in life insurance, proposing a pricing method also for liabilities which are not necessarily building blocks, but more complicated functions, like the guaranteed annuity option.

Economic Scenario Generator

The aim of this second work is to build an economic scenario generator with the intention of improving the portfolio allocation of Bpifrance.

In order to do that, we have to pass through a diﬀerent number of steps.

The first thing is to study, by a principal component analysis, the present portfolio of Bpifrance, in order to find the variables which explains the most of its variability.

A second step consists in selecting from the market the financial instruments that allows us to replicate the components we retained from the step before. This part is then completed by both an univariate and multivariate analysis of these assets, finding in this way the stylized facts that we need to take into account when choosing a model for the diﬀusion of the price of these financial factors.

The third step, and last concerning our work, is to estimate the parameters of the models we retained and see if they are able to fit the empirical data and, as a consequence, if they could be used as a part of our future economic scenario generator.

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Part I

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Introduction

Actuarial valuations involve the consideration of two type of risks: the financial risk (concerning the interest rates) and the mortality risk. Mortality is generally considered to be less diﬃcult to be mod-elled than interest rates. The traditional approach of actuaries consisted in modelling mortality in a deterministic way either assuming a suitable analytical model, either adopting adjusted or projected mortality tables. Nowadays, it is widely admitted that mortality intensities behave in a stochastic way and that they have important similarities with interest rates (see [6]). In the literature, one of the most dominant framework, when it comes to model interest rates and mortality force, is the aﬃne one, as highlighted by [19]. In such a framework one can ensure the non negativity of the quantity of interest and it admits analytical solutions for the price of financial and life insurance products traded in the market, as proposed in [6] and [15], where they exploit these advantages.

Another class of models, that can be used and allows to have closed analytical formula, is the one where the risk factor is driven by a Lévy process, as suggested in [2], where they provide a framework for the modelling of the mortality risk and the market consistent valuation of variable annuity contracts. If it is true that these models (aﬃne and Lévy) has good features (analytical tractability and market consistent valuation), it is also true that the price to pay is high in term of complexity. In fact, in order to obtain a result, one has to perform either a change of measure (which in some cases gives a new dynamics that cannot be easily computed and treated) or a Monte Carlo simulation (which can be very slow and the results depend on the number of simulations) or both.

The goal of this work is to analyse the problem of pricing life insurance liabilities using a linear rational framework: which consists on assuming the existence of a state price density ⇣ and a factor process Z with linear drift, as suggested by [19]. The main advantage of this framework is that, under some suitable assumptions, it gives us explicit and tractable formula for conditional expectation of polynomial functions of the state variable, see [18].

We assume that the factor process Z takes values in a compact state space E (as proposed in [5]). This will guarantee on the one hand the computability of the aforementioned conditional expectation, and on the other hand the possibility of using polynomial approximation in order to price life insurance contracts with more complex payoﬀ functions, such as the guaranteed annuity option for instance.

This is the main feature of our setting: there is no need to perform a Monte Carlo simulation or a change of measure if we can provide a good approximation of the conditional expectation we want to compute. This will be showed under two particular model specifications, as proposed also in [1].

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Chapter 1

Essential notions

Let us start with some fundamental tools which are essential for the understanding of the main part of this work.

In Section 1.1 we will introduce some notions about the state price density, Sections 1.2 and 1.3 state the results we are going to need about the linear rational and polynomial diﬀusion models.

1.1 State Price Density

We consider a given filtered probability space (⌦, F, F, P) such that F := (Ft)t2[0,T ], where T is a fixed

time horizon. This filtered probability space is assumed to satisfy the usual conditions: it is complete (F0contains all P-null sets) and right-continuous (Ft=Ft+=\s>tFsfor all times t).

Assumption 1.1. We can define the state price density as the strictly positive process ⇣ such that the t-price of a contingent claim with value Y in T is

⇧(t, T, Y ) =_E _⇣

T

⇣t

Y _Ft , t T.

In the case Y = 1, we obtain the price of a zero coupon bond P (t, T ) =_E

_⇣

T

⇣t Ft

, t_{ T.}

Remark 1.2. Following [38], the state price density, or also called the pricing kernel, can be defined using a reference probability Q (a risk neutral probability equivalent to P) and the inverse of the money market account Bt= exp( R

t

0rsds)(which represents the value at time 0 from investing one unit in risk

free deposits until t) as follows ⇣t= exp ✓ Z t 0 rsds ◆_d_P d_Q _F_t = exp ✓ Z t 0 rsds ◆ Vt, (1.1)

where V denotes the density process and r is an adapted process representing the short rate, such that Rt

0|rs|ds < 1 for all t 0.

Remark 1.3. It is important also to note that assuming the existence of a state price density under which the prices are martingales is a suﬃcient condition to exclude arbitrage in our market, which is the possibility of constructing a portfolio with negative price at time t = 0 and a positive payoﬀ at maturity in at least one state of the world (see [29]).

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1.2 Linear Rational Term Structure Models

A linear rational term structure model consists of two components:

i) a multivariate factor process Z with a linear drift and state space E ⇢ Rd_{. We can take the}

following dynamics (see [19])

dZt= (✓ Zt)dt + dMt,

where M is a m-dimensional martingale,  2 Rm⇥m_{and ✓ 2 R}m_.

ii) a state price density ⇣tdefined as a linear function of Zt

⇣t= ↵(t) + !(t)Zt,

for some deterministic functions ↵(t) and !(t) such that ↵(t) + !(t)z > 0 for all z 2 E.

Remark 1.4. The linear drift of the process Z implies that its conditional expectations are of the linear form (see [19]), that is

E[ZT|Ft] = ✓ + exp (T t) (Zt ✓), t T.

Lemma 1.5. The zero coupon bond prices and the short rate are linear rational functions of Zt, that is

P (t, T ) = F (T t, Zt) = ↵(T ) + ✓!(T ) + !(T )exp k(T t) (Zt ✓) ↵(t) + !(t)Zt , rt= @Tlog P (t, T )|T =t= k!(t)(Zt ✓) ↵0(t) !0(t)Zt ↵(t) + !(t)Zt .

Proof. Using Assumption 1.1, we can compute the price of a zero coupon bond as follows P (t, T ) =_E _⇣ T ⇣t Ft = E[↵(T ) + !(T )ZT|Ft] ↵(t) + !(t)Zt =↵(T ) + !(T )E[ZT|Ft] ↵(t) + !(t)Zt =↵(T ) + ✓!(T ) + !(T )exp k(T T ) (Zt ✓) ↵(t) + !(t)Zt . The last equality follows from Remark 1.4.

To compute rt, we can proceed by computing

log P (t, T ) = log ✓_{↵(T ) + ✓!(T ) + !(T )exp} _k(T _{T ) (Z} t ✓) ↵(t) + !(t)Zt ◆ , @Tlog P (t, T ) = ↵0_{(T ) + ✓!}0_{(T ) + !}0_{(T )exp} _(T _{t) (Z} t ✓) !(T )exp (T t) (Zt ✓)) (↵(t) + !(t)Zt)P (t, T ) , @Tlog P (t, T )|T =t= ↵0_{(t) + ✓!}0_{(t) + !}0_(t)(Z t ✓) !0(t)(Zt ✓) ↵(t) + !(t)Zt =!(t)(Zt ✓) ↵ 0_(t) _!0_(t)Z t ↵(t) + !(t)Zt .

1.3 Polynomial Diﬀusion Models

In this section we will give an overview of the most important results about polynomial models which will be used in our work. Let Sd _{be the space of real symmetric matrices of order d and let S}d

+ be the

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of the polynomial of degree less or equal than n on E, and by Nn the dimension of P oln(E).

We define an E-valued process Z with the following dynamics

dZt= b(Zt)dt + (Zt)dWt (1.2)

Z0= z02 E, (1.3)

where W is a d-dimensional F-adapted standard Brownian motion, and : Rd

! Rd⇥d _{is a continuous}

function such that a = 0_{, and a and b are two fixed maps such that}

a :_Rd_{! S}d aij|E2 P ol2(E) (1.4)

b :Rd _{! R}d bi|E2 P ol1(E), (1.5)

for all i and j.

Let us consider the partial diﬀerential operator G : C2₍_Rd₎_{! R associated to Z defined by}

Gf(z) =1₂tr a(z)r2_{f (z) + b(z)}0_{rf(z), z 2 R}d_.

The process Z is a polynomial diﬀusion on E if GP oln(E)✓ P oln(E)for every n 2 N.

In particular, Z is a polynomial diﬀusion if and only if conditions (1.4) and (1.5) hold (see Lemma 2.2 of [18]). In this case there exists a unique matrix representation Gn 2 RNn⇥Nn restricted to P oln(E).

In other words, for each p 2 P oln(E)with coordinate representation

p(z) = Hn(z)p, z2 Rd,

where Hn(z)is a fixed basis vector of P oln(E)and p 2 RNn, one gets

Gp(z) = Hn(z)0Gnp, z2 Rd.

The following theorems state some fundamental results about the computation of the conditional expec-tation of p(ZT).

These results are a special cases of the results of [18] taking a deterministic initial value Z0.

Theorem 1.6. F or any p 2 P oln(E)with coordinate representation p 2 RNn, we have

E[p(ZT)|Ft] = Hn(Zt)0e(T t)Gnp, t T.

Proof. See section B of [18].

The next theorem provides conditions under which the process Z admits finite exponential moments. Theorem 1.7. If ||a(z)||  C(1 + ||z||) for all z 2 E and for some constant C, then

f or any t 0, there exists ✏ > 0 such that E[e✏||Zt||_{] <}

1. Proof. See section C of [18].

The following two theorems give some suﬃcient conditions on the state space E, under which (1.2) (1.3)admits a unique weak solution.

Theorem 1.8. Let Z be an E-valued solution to (1.2) (1.3). If E is compact, then Z is weakly unique, i.e. any other E-valued solution to (1.2) (1.3), with initial value Z0, has the same law as Z.

Proof. See Theorem 5.1 of [18].

Remark 1.9. For d = 1 we have also strong uniqueness (see [5]).

Theorem 1.10. Let P be a family of polynomials on Rd_{. If the boundary of the state space E is defined}

by these polynomials, i.e. E = {z 2 Rd _{: p(z)} _{ 0 8p 2 P}, then the following conditions on a and b}

guarantee the existence of an E-valued solution to (1.2) (1.3): i) a 2 Sd

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ii) arp = 0 on {p = 0} for each p 2 P, iii) Gp > 0 on E \ {p = 0} for each p 2 P. Proof. See section E of [18].

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Chapter 2

Linear Rational Insurance Model

2.1 Financial Market

Let (⌦, F, F, P) be a filtered probability space, with P the historical measure and Ftthe economic

back-ground information available at time t. We will denote by E[·] the expectation under the measure P. We consider a framework where the prices are determined by the state price density (Assumption 1.1), which has been inspired by [19]. We suppose also a linear rational term structure, which, recalling from Section 1.2, consists of two components: a multivariate process Z with linear drift, a state space E⇢ Rm+d_{and a state price density ⇣}

tgiven as a linear function of Zt.

In particular dZt= AZtdt + dMtZ ⇣t= ↵(t) + !(t)Zt, for some i) A 2 R(m+d)⇥(m+d)_{such that A =}  c b , with c 2 Rd⇥d, b 2 Rm⇥d, 2 Rd⇥m, 2 Rm⇥m, ii) deterministic functions ↵(t) and !(t) such that ⇣t> 0for every z 2 E,

iii) (m + d)-dimensional martingale MZ_.

Remark 2.1. The Ft-conditional expectation of ZT is linear in Zt,

E[ZT|Ft] = exp A(T t) Zt, t T. (2.1)

This remark is a generalisation of Remark 1.4.

Lemma 2.2. The price of a zero coupon bond is given by the following closed formula P (t, T ) = F (T t, Zt) =

↵(T ) + !(T )exp A(T t) Zt

↵(t) + !(t)Zt

. (2.2) Proof. See proof of Lemma 1.5.

2.2 Mortality Model

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We consider a survival process S of an insured person, which is a right-continuous F-adapted and non-increasing strictly positive process, with S0= 1.

Let U ⇠ U([0, 1]) independent from FT. We can define the random death time of a policyholder in this

way

⌧ = inf_{t 0 : St U}, (2.3)

which is infinite if the set is empty.

Let D be the filtration generated by the indicator process D defined by Dt= 1{⌧ t}.

The random time ⌧ is a stopping time in the enlarged filtration F _ D and it is doubly stochastic (see [1]), in the sense that

P[⌧ > t|FT] =P[St> U|FT] = St.

The filtration G = F _ D contains all the economic and mortality information.

In our study we suppose that the survival process is modelled by St = a0Yt for some a 2 Rd+ s.t.

a01 = 1and the process Y is defined below.

Assumption 2.3. The process Z will be decomposed into two separate processes (Y, X) with values in Rd

+⇥ Rmwith linear drift of the form

dYt= (cYt+ Xt)dt + dMtY (2.4)

dXt= (bYt+ Xt)dt + dMtX, (2.5)

for some c 2 Rd⇥d_{, b 2 R}m⇥d_, _{2 R}d⇥m_, _{2 R}m⇥m_{, m-dimensional F}

t-martingale MtX and

d-dimensional Ft-martingale MtY.

The process S being positive and non-increasing, we necessarily have that its martingale component MS

t = a0MtY is of finite variation and thus purely discontinuous, and that St < MtS  0 because

St= MtS. This is the reason of the decomposition of Z into a component X and a component Y of

finite variation (see [1]).

We will make use of the following results

Lemma 2.4. Let Y be a non negative FT-measurable random variable. For any t  tM  T ,

E[1{⌧>tM}Y|Gt] = 1{⌧>t}

1 StE[S

tMY|Ft].

Proof. See Lemma A.1 of [1].

Lemma 2.5. Let Z be a bounded F-predictable process. For any t  tM < T,

E[1{t<⌧tM}Z⌧|Gt] = 1_{⌧>t} St E  Z tM t ZudSu Ft .

Proof. See Lemma A.2 of [1].

Proposition 2.6. The process S is non increasing if and only if the following conditions are satisfied i) a0 _{< M}Y _{>= 0}_{almost surely, for the continuous part of the martingale M}Y_,

ii) a0_(cY

t+ Xt) 0 almost surely for almost every t,

iii) a0 _MY

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Proof. To prove this proposition we can compute the variation of S, St= a0 Yt

= a0(cYt+ Xt) t + a0 MtY

 0

The inequality follows from the conditions i, ii and iii of the proposition. This prove the suﬃciency of this conditions.

For the necessary part: if S is decreasing for every t, this means that S is a finite variation process (condition i). In addition, since S is decreasing, its drift and its jumps have to be negative (conditions iiand iii). This concludes our proof.

In addition, we will suppose the following covariances between our variables d < Yi_{, Y}j_>

t= ⌫1,yij Yt+ ⌫2,yij (Yt)2+ 1,yij Xt+ 2,yij (Xt)2+ ⌘ijy dt, i, j = 1, . . . , d (2.6)

d < Xi, Xj>t= ⌫1,xij Yt+ ⌫2,xij (Yt)2+ ij1,xXt+ 2,xij (Xt)2+ ⌘xij dt, i, j = 1, . . . , m (2.7)

d < Yi, Xj>t= ⌫1,xyij Yt+ ⌫2,xyij (Yt)2+ ij1,xyXt+ ij2,xy(Xt)2+ ⌘ijxy dt, i = 1, . . . d, j = 1, . . . , m,

(2.8) where (Yt)2and (Xt)2 denote the vectors Ytand Xtwhere all their components are taken to the second

power, ⌫ij _{is a d dimensional vector,} ij _{is a m dimensional vector and ⌘}ij _{is a real number.}

Remark 2.7. The information about the covariances is contained in the following matrices, < Y >t 2 Rd⇥d,

< X >t 2 Rm⇥m,

< Y, X >t2 Rd⇥m.

Lemma 2.8. If S is absolutely continuous (i.e. a0 _MY

t = 0) then, the mortality intensity µt, which is

derived from the relation St= exp( R₀tµsds), is linear rational in (Yt, Xt)and takes this expression,

µt= a 0_(cY

t+ Xt)

a0_Y_t . (2.9)

Proof. Starting from St = exp( R t

0µsds) and taking the derivative with respect to the variable t, we

obtain St( µt)dt = dSt, which leads us to St( µt) = a0(cYt+ Xt) µt= a0(cYt+ Xt) a0_Y_t .

2.2.1 Modelling of the Mortality Force

In this section we will show how we can choose an appropriate linear rational model in order to take into account diﬀerent demographic features of the mortality intensity.

According to [35] and [44] we can observe diﬀerent trends in mortality, one among them is the so called rectangularization phenomenon:

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range, and the curve starts to appear more rectangular. According to [21], this phenomenon, which can be observed in industrialised countries, suggests that human life expectancy is approaching its maximum potential value.

It is important to note also that the rectangularization may depend on the age range and the time frame of analysis. Actually, the variability of age at death may decline in some periods when the entire population is considered, but may remain stable if the analysis is limited to older ages ([37]). Moreover, there is no reason to expect that any process of compression of mortality should continue indefinitely. Another important aspect to take into account is that the average length of life (the life expectancy at birth) is in general independent of its variability. Therefore limits to life expectancy and the rectan-gularization phenomenon should be considered as two diﬀerent issues. By the way, in some particular cases, [21] arguments that these two concepts may be interrelated. He establishes an upper limit of the life expectancy and believes that the maximum human life span is fixed. This is a very important statement, because if it were true, then it would mean that the distribution of ages at death is bounded on the right with the consequence that if mortality declines and death is delayed, the distribution of the ages at death must be compressed and the survival function must become more rectangular. Despite this statement seems to be quite logical, [37], [42] and [45] showed that from an empirical point of view the rectangularization has not continued on recent decades for elderly population and the proportion of centenarians in a population has not been constant and unchanging over time. Even if those studies argue that rectangularization is not necessarily associated with the mortality decline, the connection between these two aspects remains indeterminate ([43]).

Another important trend that can be derived from [44] concerns the fact that the variability of age at death has decreased during the past 250 years. This reflects three important phases of mortality decline:

i) period of slow mortality reduction across the age range, associated with high but stable levels of variability in age of death.

ii) an era of rapid reduction in infant and child mortality, resulting in an enormous and rapid com-pression of mortality.

iii) a period of unprecedented reduction in late-adult mortality, followed by near-constant levels of variability in ages at death.

According to [44], the current pattern of mortality suggests that the level of variability observed today could be maintained for the foreseeable future.

This aspects give rise to a reduction of the risk of mortality and an augmentation of the risk of longevity (the risk that a person lives beyond its life expectancy), see [6].

So, when choosing a model, we need to take into account this new kind of risk.

The first thing to do, as suggested in [6], is to calculate some indicators that could help to understand the shape of the survival and death curves. We can compute:

i) the entropy of the survival function: H = R! 0 log(S_R t)/Stdt ! 0 Stdt ,

where ! is the maximal age of a policyholder. This indicator measures the rectangularization of the survival function implied by the mortality force.

If H ! 0 it means that the shape of the survival function tends to be rectangular. ii) the expectation of life for a policyholder of age x:

E[⌧x] =

Z ! x 0

Stdt.

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It is known (see [6]) that processes with non time dependent coeﬃcient can hardly capture these demo-graphic features we stated. That is why it would be interesting to propose some special cases of our framework.

We present now two examples of how we can integrate these empirical evidences (the longevity risk and the rectangularization phenomenon) in our linear rational framework. This can be done either modify-ing the survival process S or the process (Y, X). These time dependent coeﬃcients could represent for instance an available mortality table, a prudential demographic basis when carrying out risk analysis or a realistic assumption on µ (see [6]).

Examples

1) We can treat the case where the coeﬃcients of the survival process are time dependent, i.e. St= a0(t)Yt, such that a0(t)1 = 1.

Lemma 2.9. In this case the mortality intensity takes the following expression µt= a

0_(t)Y

t+ a(t)(cYt+ Xt)

a0_(t)Y_t . (2.10)

Proof. From the application of the Itô’s lemma on the process S = a0_(t)Y_{, we have that}

dSt= a0(t)Ytdt + a0(t)dYt. Then, by using Lemma 2.8, we get

dSt= µtStdt

µt= a 0_(t)Y

t+ a0(t)(cYt+ Xt)

a0_(t)Y_t .

2) Another interesting case would be the one with time dependent coeﬃcients in the dynamics of (Y, X), that is

dYt= c(t)Yt+ (t)Xt dt + dMtY (2.11)

dXt= b(t)Yt+ (t)Xt dt + dMtX. (2.12)

Remark 2.10. In this case equation (2.1) becomes E   YT XT Ft = exp ✓ Z T t A(u)du ◆  Yt Xt . (2.13)

Lemma 2.11. The new mortality force is defined as µt=

a0 _c(t)Y

t+ (t)Xt

a0_Y_t . (2.14)

Proof. See Lemma 2.8.

In this work we will principally focus on the case of non time dependent coeﬃcients, nevertheless we will extend our results to these two special cases whenever this would lead to a diﬀerent or interesting result.

2.3 Valuation of insurance contracts

In this section we will evaluate the survival and death benefits of standard insurance contracts under the assumptions of model (2.4) (2.5).

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Assumption 2.12. The arbitrage-free prices of GT-measurable claims are generated by the same state

price density ⇣ of the claims that are FT-measurable, which implies that this state price density will

satisfy the same stochastic diﬀerential equation, i.e. dynamics, in G.

Before presenting the main result, we will introduce some important notation. Definition 2.13. We will note:

i) Hn(y) =



y1y2 . . . (y1)n (y2)n . . . is a polynomial basis of P oln(Rd+)and it will be also denoted

by Yn.

ii) H⇤

n(y, x) =



x1 x2 . . . x21x1y x22 x2y . . . xm . . . is a vector which contains only the polynomials

of order n which are function of x and xy. It will be also denoted by Xn.

iii) Hn(y, x) =

✓

Hn(y), Hn⇤(y, x)

◆

is a polynomial basis of the whole space E. iv) n 2 N+_.

The following theorem state an important result, which will allow us to price life insurance contracts even when their payoﬀs are not a linear function of the factor process (Y, X), but a polynomial. The idea is to consider this polynomial as a linear function of an enlarged set of variables, which will enable us, under our assumptions, to use all the results we have concerning linear drift processes.

Theorem 2.14. Let us consider the process (Y, X) as defined in (2.4) (2.5), then the process Hn(Y, X) =

(_Yn,Xn) has linear drift in Yt,n and Xt,n.

An analogous general result can be found in [18].

For brevity of notation we will not indicate explicitly the order of the polynomials (as we assumed that it is n) and the dependence on time in the gradient and the Hessian matrix.

Proof. To prove this theorem we will proceed by computing the dynamics of Y and of X .

In order to compute the entire dynamics of the process, it will be suﬃcient to calculate the dynamics of their generic components.

i) we know that Y = Hn(Y ) =



Y1 Y2 . . . Y1nY2n . . . , we need to show that the drift of the generic

component h is linear in Yt and Xt.

So we take Yh_{= (Y}1₎q1_{. . . (Y}d₎qd, such thatPd

i=1qi  n and qi 0 f or any i = 1, . . . , d.

Then, by application of Itô’s lemma, we have d_Yth=r0h,ydYt+

1

2tr Hh,yd < Y >t ,

where rh,yand Hh,yare respectively the gradient and the Hessian matrix of Yh(taken with respect

to the variable y), and tr(·) is the trace operator. In particular they are equal to

rh,y= ✓ qi(Yti)qi 1 Y j6=i (Ytj)qj ◆ i=1,...,d Hh,y= ✓ qi(qi 1)(Yti)qi 21{i=j} Y k6=i (Ytk)qk+ (qiqj(Yti)qi 1(Y j t)qj 11{i6=j} Y k6=i,j (Ytk)qk ◆ i,j=1,...d . Then, replacing the dynamics of Y , we have

dYh t =r0h,y(cYt+ Xt)dt + 1 2tr(Hh,yd < Y >t) +r 0 h,ydMtY

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where

Y_(Y t) =

1

2tr Hh,yd < Y >t .

By regrouping the dt-terms and observing that they are a linear transformation of Ytand Xt, one

has

d_Yth= (˜chYt+ ˜hXt)dt +r0h,ydMtY,

where ˜ch is a row vector of dimension dY = dim(Y) and ˜h is a another row vector of dimension

dX = dim(X ) such that they solve the following equation ˜

chYt+ ˜hXt=r0h,ycYt+r0h,y Xt+ Y(Yt).

Finally, combining the dynamics of all the components of Y, we obtain

d_Yt= (˜cYt+ ˜Xt)dt +r0ydMtY, (2.15)

where ˜c 2 RdY⇥dY

, ˜ 2 RdY⇥dX

and r0

y is the matrix of dimension dY ⇥ d which contains the

transposed gradients of every component of Y.

ii) We know that X = H⇤

n(Y, X) =



X1 . . . Xm . . . X1Y1 . . . .

As before, we take the generic component h, Xh _{= (Y}1₎p1_{. . . (X}m₎pd+m, such that Pd+m

i=1 pi  n, pi 0 for i = 1, . . . , d and pi > 0 for

i = d + 1, . . . , m.

By Itô’s lemma, we have

d_Xth=r0h,x  dYt dXt + 1 2tr Hh,xd < Y, X >t ,

where rh,x is the gradient of Xth and Hh,x is the Hessian matrix of Xh (both taken with respect

to the variables x and y), which are defined as follows rh,x= ✓ pi(Zti)pi 1 Y j6=i (Ztj)pj ◆ i=1,...,d+m Hh,x= ✓ pi(pi 1)(Zti)pi 21{i=j} Y k6=i (Ztk)pk+ (pipj(Zti)pi 1(Z j t)pj 11{i6=j} Y k6=i,j (Ztk)pk ◆ i,j=1,...d+m , where Zj t = ( Ytj if j  d Xtj dotherwise . Then, we get d_Xth=r0h,x  cYt+ Xt bYt+ Xt dt + 1 2tr Hh,xd < Y, X >t +r 0 h,x  dMY t dMX t = ✓ r0h,x  cYt+ Xt bYt+ Xt + XY_(Y t, Xt) ◆ dt +_r0h,x  dMY t dMX t , where XY_(Y t, Xt) = 1 2tr Hh,xd < Y, X >t .

Now, we can observe that the dt-terms are a linear transformation of Ytand Xtand obtain

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where ˜bh is a row vector of dimension dY and ˜h is another row vector of size dX, which solve the following equation ˜bhYt+ ˜hXt=r0h,x  cYt+ Xt bYt+ Xt + 1 2tr Hh,xd < Y, X >t . Finally, the dynamics of X is

dXt= (˜bYt+ ˜Xt)dt +r0x  dMtY dMX t , (2.16) with ˜b 2 RdX_⇥dY , ˜ 2 RdX_⇥dX and r0

xis the matrix of dimension dX⇥ (d + m) which contains the

transposed gradients of all components of Xh_.

The dynamics of the process (Y, X ) is

d_Yt= (˜cYt+ ˜Xt)dt +r0ydMtY d_Xt= (˜bYt+ ˜Xt)dt +r0x  dMY t dMX t .

Lemma 2.15. Let (Y, X ) be a process defined by (2.15) (2.16), then ifRt

0r0ydMsY and Rt 0r0x  dMsY dMX s

are martingale for t  T , one has E   YT XT Ft = exp ˜A(T t)  Yt Xt , t T. (2.17) where ˜A =  ˜ c ˜ ˜b ˜

Proof. The proof follows by Remark 2.1 and by linearity of the drift of our new process (Y, X ).

Conditions such that the martingale condition is verified will be specified later in some particular cases.

Remark 2.16. It is important to notice that in the case where the process (Y, X) has time dependent coeﬃcients, assumed in (2.11) (2.12), the result of Theorem 2.14 is still valid, in particular

dYt= ˜c(t)Yt+ ˜(t)Xt dt +r0ydMtY d_Xt= ˜b(t)Yt+ ˜(t)Xt dt +r0x  dMY t dMX t , and, provided that the condition stated in Lemma 2.15 is satisfied,

E   YT XT Ft = exp ✓ Z t 0 ˜ A(u)du ◆  Yt Xt , t T.

From now on we will focus on the pricing of some life insurance liabilities, which can be considered as building blocks for other products, using the model we specified.

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2.3.1 Survival benefit

A survival benefit is the benefit CT, F-measurable, given to the policyholder at the end of his policy

tenure (the time T ).

The value at time t  T of this insurance contract is given by SBt(CT, T ) =E _⇣ T ⇣t 1_{{⌧>T }}CT Gt = 1_{⌧>t} ⇣tSt E ⇥ ⇣T1{⌧>T }CT Ft⇤ =1{⌧>t} ⇣tSt E ⇥ ⇣TE[1{⌧>T }|FT]CT Ft⇤= 1_{⌧>t} ⇣tSt E ⇥ ⇣TSTCT Ft⇤, (2.18)

where ⌧ is the random death time (see formula (2.3)). The second equality follows from Lemma 2.4. In the following sections we will treat two diﬀerent cases: the one where the benefit is constant (insurances with amount fixed at the start of the contract, see [5]), and the one where the benefit is a polynomial function of order k of Y .

The interest in considering constant benefits, besides its simplicity from a mathematical point of view, lies in the fact that this kind of contracts are actually traded on the market and they are preferred to the ones with time dependent or random benefits by the average policyholder. An example is a term life insurance which pays out a fixed amount of money to the beneficiary in case of death.

A) CT = C2 R+

Using the representation of ⇣T and ST, the conditional expectation in (2.18) becomes

E⇥⇣TST Ft⇤=E ✓ ↵(T ) + !(T )  YT XT ◆ a0YT Ft =E⇥↵(T )a0YT + !1(T )YTa0YT + !2(T )XTa0YT Ft⇤,

where !1(·) is the vector consisting of the first d components of !(·) ,and !2(·) is the one containing

the last m components of !(·).

Then, applying Theorem 2.14 and by linearity, we have E⇥⇣TST Ft⇤= a0s(T )exp ˜A2(T t)

 Yt,2

Xt,2 , (2.19)

where as(·) is such that

a0s(T )H2(y, x) = !1(T )ya0y + ↵(T )a0y + !2(T )xa0y

where H2(y, x)is a polynomial basis of the state space P ol2(Rd+⇥ Rm+)(see Definition 2.13).

B) CT = a0cHk(YT)

We can extend (2.19) to the case CT = a0cHk(YT).

The computations are the same, only the dimension of the problem increases, going from 2 to 2+k, as we will treat polynomials of degree k + 2 instead of polynomial of degree 2.

Proposition 2.18. The value of the survival benefit in the hypothesis CT = a0cHk(YT)is

SBt(CT, T ) = 1_{⌧>t} St⇣t a0_s(T )exp ˜Ak+2_(T _t)  Yt,k+2 Xt,k+2 . (2.20)

where as(·) is such that

a0s(T )Hk+2(y, x) = ↵(T )a0cHk(y)a0y + !1(T )ya0cHk(y)a0y + !2(T )xa0cHk(y)a0y.

Proof. Starting from the conditional expectation in (2.18) and using the expression of CT and ⇣T,

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Remark 2.19. In the case where the survival process S or the process (Y, X) have time dependent coeﬃcients, the prices found in (2.19) and (2.20) are not valid anymore. Anyway, we are always able to compute the price of the survival benefit under the hypothesis that the benefit C is a polynomial of order k of the process Y , i.e. CT = a0cHk(YT).

1) St= a(t)Yt. In this case the expectation in (2.18) changes into

E⇥⇣TSTa0cHk(YT)Ft⇤=E ✓ ↵(T ) + !(T )  YT XT ◆ a0(T )YTa0cHk(YT)Ft =_E⇥a0s(T )Hk+2(YT, XT)Ft⇤ = a0s(T )exp ˜Ak+2(T t)  Yt,k+2 Xt,k+2, ,

where as(·) satisfies the following equation

a0s(T )Hk+2(y, x) = ↵(T ) + !1(T )y + !2(T )x a0(T )ya0cHk(y)

2) (Y, X) has time dependent coeﬃcients. Under this assumption, the expected value in (2.18) becomes E⇥⇣TSTa0cHk(YT)Ft⇤=E ✓ ↵(T ) + !(T )  YT XT ◆ a0YTa0cHk(YT)Ft =_E⇥a0s(T )Hk+2(YT, XT)Ft⇤ = a0_s(T )exp ✓ Z T t ˜ Ak+2_(u)du ◆ du  Yt,k+2 Xt,k+2 , where as(·) satisfies a0s(T )Hk+2(y, x) = ↵(T ) + !1(T )y + !2(T )x a0ya0cHk(y) .

2.3.2 Death benefit

A death benefit is a payout C⌧ to the beneficiary of a life insurance policy, annuity, or pension when the

insured or annuitant dies (before the time T ).

Assumption 2.20. The process S is assumed to be almost surely continuous.

This means we exclude the possibility that in a given population the mortality could change due to an instantaneous shock.

The value at time t  T of the death benefit is given by DBt(C, T ) =E _⇣ ⌧ ⇣t 1_{{t<⌧T }}C⌧ Gt = 1_{⌧>t} ⇣tSt E  Z T t ⇣uCudSu Ft = 1{⌧>t} ⇣tSt E  Z T t ⇣uCu[ a0(cYu+ Xu)]duFt = 1{⌧>t} ⇣tSt Z T t E ⇥ ⇣uCu[ a0(cYu+ Xu)]Ft⇤du. (2.21)

The second equality follows from Lemma 2.5 and the last inequality follows from the application of Fubini-Tonelli Theorem, which is possible thanks to the positivity of the integrand.

As before we will treat the case where C⌧ is a constant and the case where C⌧ is a polynomial function

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A) C⌧ = C2 R+

Following the same procedure as before, the integrand in (2.21) can be rewritten as E ✓ ↵(u) + !(u)  Yu Xu ◆_⇥ a0(cYu+ Xu)⇤ Ft

=E⇥ ↵(u)a0(cYu+ Xu) !1(u)Yua0(cYu+ Xu) !2(u)Xua0(CYu+ Xu)Ft⇤

= a0d(u)exp ˜A2(u t)

 Yt,2

Xt,2 , (2.22)

where ad(·)is such that

a0d(u)H2(y, x) = !1(u)ya0cy ↵(u)a0cy !1(u)ya0cx !2(u)xa0(cy + x) ↵(u)a0 x.

B) C⌧ = a0cHk(Y⌧)

The same procedure can be used to compute the death benefit under the hypothesis C⌧= a0cHk(Y⌧).

Proposition 2.21. The price of the death benefit when C⌧ = a0cHk(Y⌧)is

DBt(C⌧, T ) = 1_{⌧>t} St⇣t Z T t ✓ a0d(u)exp ˜Ak+2(u t)  Yt,k+2 Xt,k+2 ◆ du, (2.23) where a0

d(·) satisfies the following equation

a0d(u)Hk+2(y, x) = ↵(u) + !1(u)y + !2(u)x a0cHk(y)a0(cy + x).

Proof. From (2.21), focusing on the conditional expectation and using the hypothesis that C⌧ is a

polynomial, we have

E⇥⇣uCu[ a0(cYu+ Xu)]Ft⇤=E⇥ ⇣ua0cHk(Yu)a0(cYu+ Xu)Ft⇤

=E⇥ (↵(u) + !1(u)Yu+ !2(u)Xu)a0cHk(Yu)a0(cYu+ Xu)Ft⇤

= a0d(u)exp ˜A2(u t)

 Yt,k+2

Xt,k+2 ,

Remark 2.22. As we did in the last section, we can study the price of the death benefit under the time dependent coeﬃcients framework, when the benefit C⌧ is a polynomial function of order k of Y , that is

C⌧= a0cHk(Y⌧)

1) St= a(t)Yt. The integrand of (2.21) becomes

E⇥⇣ua0cHk(Yu)[ a0(u)(cYu+ Xu)]Ft⇤=E  ✓ ↵(u) + !(u)  Yu Xu ◆ a0cHk(Yu)a0(u)(cYu+ Xu) =E⇥a0d(u)Hk+2(Yu, Xu)Ft ⇤ = a0d(u)exp ˜Ak+2(u t)  Yu,k+2 Xu,k+2, , where ad(·) satisfies

a0d(u)Hk+2(y, x) = ↵(u) + !1(u)y + !2(u)x a0cHk(y)a0(u)(cy + x).

2) (Y, X) has time dependent coeﬃcients. In this case, the expectation in (2.21) changes into E⇥⇣ua0cHk(Yu)[ a0(cYu+ Xu)]Ft ⇤ =E  ✓ ↵(u) + !(u)  Yu Xu ◆ a0cHk(Yu)a0(cYu+ Xu) =E⇥a0d(u)Hk+2(Yu, Xu)Ft⇤ = a0d(u)exp ✓ Z u t ˜ Ak+2(s)ds ◆  Yu,k+2 Xu,k+2 , where ad(·) is solution of

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2.3.3 Whole Life Annuity

Another life insurance contract which can be priced under our framework is the Whole Life Annuity. This contract allows the insured to get a periodic payoﬀ until his death. This kind of insurance products are usually purchased by investors who want to secure an income stream during retirement.

If we suppose ! to be the maximal age of the policyholder and x his age at time t, then the whole life annuity can be seen as a sum of ! x survival benefits which are dependent on C, which is the periodic payoﬀ.

Proposition 2.23. The price at time t  T of a whole life annuity, which gives the right to the insured to get a sum Cj until his death, is

ax(t, C) = ! x_X

j=0

SBt(Ct+j, t + j). (2.24)

Then, under the hypothesis Cj= a0cHk(Yj), (2.24) can be rewritten as

ax(t, C) = ! x_X j=0 1_{⌧>t} St⇣t a0s(j)exp ˜Ak+2(j t)  Yt,k+2 Xt,k+2 . (2.25)

Proof. This is obtained by applying the result in Proposition 2.18 (which gives us the t-value of survival benefit when C is a polynomial function of Y ) to (2.24).

Remark 2.24. We will denote by ax(t) the t-price of a whole life annuity which pays a unit of money

until the death time of the insured (see Section 3.1 of [15]).

2.3.4 Guaranteed annuity option

The guaranteed annuity option (GAO) is the contract who gives to the insured the right to choose, at time T , between an annual payment g until his death time, where g is the fixed rate called the guaranteed annuity rate, or a cash payment equal to the capital 1.

The value at time T of this contract is

V (T ) = max gax(T ), 1 = 1 + g max ✓ ax(T ) 1 g, 0 ◆ . (2.26) Focusing on the second term, which is called the GAO option price entered by an x-year policyholder at time t, the value at time t  T will be

GAOx(t, T ) =E _⇣ T ⇣t g max ✓ ax(T ) 1 g, 0 ◆ Ft . (2.27)

This payoﬀ is very similar to the one of a basket option.

A way to treat this value is by approximation. The idea is to approximate the payoﬀ function by a polynomial and then apply Theorem 2.14 to compute the expectation and obtain a closed formula for the price of this payoﬀ.

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Polynomial approximation

In this section we present the method we will apply to approximate the payoﬀ function f (XT, YT) = ✓ ax(T ) 1 g ◆+ (2.28) by a polynomial.

To do that we will focus on the method proposed in [5] and used also in [19]. Let us consider a Ft-conditional expectation of the form

⇠(t, T ) =_E⇥f (YT, XT)Ft⇤,

for some continuous function f(y, x) on the state space E.

We will make use of the following assumption on the state space E Assumption 2.25. The state space E ⇢ Rd _{is assumed to be compact.}

This assumption provides that we can always find a polynomial approximation {fm}m2Nof f on E

of order n 2 N, i.e. lim m!1||fm f||1= 0, where ||f||1= sup x2E|f(YT , XT)|,

for any f 2 C(E) (see [5]).

Lemma 2.26. Let (fm)m2N be a uniform polynomial approximation of the continuous function f on E.

Then, we consider for every m 2 N

GAOx(t, T )m= gE

_⇣

T

⇣t

fm(YT, XT)Ft .

{GAOx(t, T )m}m2N provides both a pathwise and Lp approximation of GAOx(t, T ) in (2.27) for any

p 1 uniformly in t 2 [0, T ], i.e. sup t2[0,T ]|GAO x(t, T )m GAOx(t, T )| m!1! 0 a.s. sup t2[0,T ]||GAO

x(t, T )m GAOx(t, T )||Lp m!1! 0 for every p 1.

Proof. See Lemma 4.2 of [5].

We are now able to compute the price of our approximated payoﬀ GAOx(t, T )m= gE _⇣ T ⇣tfm(YT, XT)Ft = g ⇣tE ⇥ ⇣Tfm(YT, XT)Ft⇤ = g ⇣tE ✓ ↵(T ) + !(T )  YT XT ◆ fm(YT, XT)Ft = g ⇣tE ⇥ a0g(T )Hn+1(YT, XT)Ft⇤ = g ⇣t a0g(T )exp ˜An+1(T t)  Yt,n+1 Xt,n+1 . (2.29)

The last equality is obtained by applying Lemma 2.15.

The deterministic function ag(·) satisfies the following equation

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Remark 2.27. Like the survival and the death benefit, we can compute the price of the GAO also when the process (Y, X) has time dependent coeﬃcients. We can notice also that the case St= a(t)Ytdoes not

change the value of (2.29), as the approximated payoﬀ does not depend on the survival process S. Then, the price of the guaranteed annuity option turns into

GAOx(t, T )m= gE  ⇣T ⇣t fm(YT, XT)Ft = g ⇣tE ⇥ ⇣Tfm(YT, XT)Ft⇤ = g ⇣tE ✓ ↵(T ) + !(T )  YT XT ◆ fm(YT, XT)Ft = g ⇣tE ⇥ a0g(T )Hn+1(YT, XT)Ft⇤ = g ⇣t a0g(T )exp ✓ Z T t ˜ An+1(u)du ◆  Yt,n+1 Xt,n+1 ,

where ag(·) solve the following equation

a0_g(T )Hn+1(y, x) = ↵(T ) + !1(T )y + !2(T )x fm(y, x).

2.4 Application

We will focus now on some applications of the linear models in order to price the life insurance contracts we presented in the previous sections. We will treat the Linear Hypercube model (LHC) and in particular two special cases: the one-factor LHC and the two-factors cascade LHC (studied in [1]). This class of models is appropriate for the modelisation of the mortality, as they allow the mortality intensity to be a non negative process.

2.4.1 The Linear Hypercube Model

The Linear Hypercube Model is a model with d = 1 (so that St = Yt), where the survival process is

absolutely continuous and the factor process X is diﬀusive and takes values in a hypercube whose edges’ length is given by Y (see [1]),

E =_{{(y, x) 2 R}1+m: y_{2 (0, 1] and x 2 [0, y]}m_}. (2.30) The dynamics of (Y, X) is

dYt= 0Xtdt (2.31)

dXt= (bYt+ Xt)dt + ⌃(Yt, Xt)dWt, (2.32)

for some 2 Rm

+ and some m-dimensional brownian motion W , where the volatility matrix is given by

⌃(y, x) = diag 1

p

x1(y x1), ..., m

p

xm(y xm) ,

with volatility parameters 1, ..., m 0.

Remark 2.28. The mortality intensity, using the relation St = Yt, can be expressed in the following

way µt= 0Xt Yt . In addition, 0_{ µ}t 01, see [1].

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Proposition 2.29. If, for all i = 1, . . . , m, bi X ji ij 0 i+ ii+ bi+ X ji ( j+ ij)+ 0,

then, for any initial value (X0, Y0) with support E, there exists a unique E-valued solution (Y, X) of

(2.31) (2.32). It satisfies the boundary non-attainment, for some i = 1, . . . , m, i) Xit> 0 for all t 0if Xi0> 0 and bi Pji ij

2 i

2

ii) Xit< Yit for all t 0 if Xi0< Yi0 and i+ ii+ bi+Pji( j+ ij)+

2 i

2

Proof. See proof of Theorem 3.1 of [1].

Remark 2.30. It is important to note that the martingale condition stated in Lemma 2.15 is automati-cally satisfied, this because of the compactness of the state space E (see equation (2.30)), which enables us to use the result proved in the abovementioned lemma.

2.4.2 LHC(1)

Let us consider the Linear Hypercube model with m = 1. Then, the dynamics of the process (Y, X) is dYt= Xtdt

dXt= (bYt+ Xt)dt +

p

Xt(Yt Xt)dWt,

E =_{{(y, x) 2 R}2: y_{2 (0, 1] and x 2 [0, y]}.} We recall that St= Yt.

Remark 2.31. The conditions stated in Proposition 2.29 can be rewritten as b 0, ( + b + ) 0.

We are interested in the computation of the price of the survival, death benefit and the guaranteed annuity option.

In particular, for the survival and death benefit we will focus on the case where the benefit is a positive constant C.

Survival and Death Benefit

From Sections 2.3.1 and 2.3.2, we need to define a new process (Y2,X2), where Y2= H2(Y ) =

 Y Y2

and X2= H2⇤(Y, X) =



X X2 _XY _.

Proposition 2.32. The dynamics of (Y2,X2) is

d_Yt,2= ˜0Xt,2dt (2.33)

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where ˜0=  0 0 0 0 2 , ˜b = 2 4b0 00 0 b 3 5 , ˜ = 2 40 2 0 2 _{2b +}0 2 0 3 5 , ˜ ⌃(_Yt,2,Xt,2) = 2 4 p Xt(Yt Xt) 2 XtpXt(Yt Xt) Yt p Xt(Yt Xt) 3 5 .

Proof. To prove our statement we can use Theorem 2.14 and we get i) for Y2 we have dYt,2= ( dYt 2YtdYt = ( Xtdt 2 YtXtdt = ( ˜10Xt,2dt ˜0 2Xt,2dt = ˜Xt,2dt.

ii) for X2 we have

d_Xt,2= 8 > < > : dXt 2XtdXt+ d < X >t YtdXt+ XtdYt = 8 > < > : (bYt+ Xt)dt + pXt(Yt Xt)dWt (2bXtYt+ 2 Xt2)dt + 2 Xt p Xt(Yt Xt)dWt+ ( 2XtYt 2Xt2)dt (bY2 t + XtYt)dt + Yt p Xt(Yt Xt)dWt Xt2dt = 8 > < > : (˜b1Yt,2+ ˜1Xt,2)dt + ˜⌃1(Yt,2,Xt,2)dWt (˜b2Yt,2+ ˜2Xt,2)dt + ˜⌃2(Yt,2,Xt,2)dWt (˜b3Yt,2+ ˜3Xt,2)dt + ˜⌃3(Yt,2,Xt,2)dWt = (˜bYt,2+ ˜Xt,2)dt + ˜⌃(Yt,2,Xt,2)dWt.

Now, the t-price of the survival benefit is SBt(C, T ) = C1{⌧>t} ⇣t E ⇥ ↵(T )YT+ !1(T )YT2+ !2(T )XTYT Ft⇤ = C1{⌧>t} ⇣t a 0 s(T )exp ˜A2(T t)  Yt,2 Xt,2 , where a0 s(T ) = [↵(T ) !1(T ) 0 0 !2(T )]and ˜A2= ₀ _˜ ˜b ˜ , and t-price of the death benefit is

DBt(C, T ) = C1{⌧>t} ⇣t Z T t E ⇥

↵(u)Xu+ !1(u)XuYu+ !2(u)Xu2 Ft⇤du

= C1{⌧>t} ⇣t Z T t a0d(u)exp ˜A2(T t)  Yt,2 Xt,2 du, where a0

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Guaranteed annuity option

For what concerns the GAO price, the approximation given by (2.29) is a polynomial of degree n + 1. Without loss of generality we present the computations for the case of a polynomial of degree n. So, we need to define a new process (Yn,Xn), where

Yn= Hn(Y ) =  Y Y2 . . . Yn , Xn= Hn⇤(Y, X) =  X X2_{XY X}3 _X2_{Y XY}2 _{. . . X}n _{. . . XY}n 1 _.

Proposition 2.33. The dynamics of (Yn,Xn)is

d_Yt,n= ˜0Xt,n (2.35) d_Xt,n= (˜bYt,n+ ˜Xt,n)dt + ˜⌃(Yt,n,Xt,n)dWt. (2.36) where ˜ 2 RdY⇥dX , ˜b 2 RdX⇥dY , ˜ 2 RdX⇥dX and ˜⌃ 2 RdX_.

We recall that dY _{= dim(}_Y

t,n)and dX = dim(Xt,n). ˜0= 2 6 6 6 4 0 0 . . . 0 . . . 0 0 0 2 . . . 0 . . . 0 ... ... ... ... ... ... ... 0 0 0 . . . n . . . 0 3 7 7 7 5, ˜b = 2 6 6 6 6 6 4 b 0 0 . . . 0 0 0 0 . . . 0 0 b 0 . . . 0 ... ... ... ... ... 0 0 0 . . . b 3 7 7 7 7 7 5 , ˜ = 2 6 6 6 6 6 6 6 6 6 6 6 4 0 0 0 0 0 . . . 0 0 0 2 2 2b + 2 0 0 0 . . . 0 0 0 0 0 0 . . . 0 0 0 0 0 3 3 2 _{3b +} 2 ₀ _{. . .} ₀ ₀ 0 0 0 2 2 _{2b +} 2 _{. . .} ₀ ₀ 0 0 0 0 2 . . . 0 0 ... . . ... ... . . . 0 0 0 0 0 0 0 0 . . . (n 1) 3 7 7 7 7 7 7 7 7 7 7 7 5 , ˜ ⌃(_Yt,n,Xt,n) = 2 6 4 p Xt(Yt Xt) ... Ytn 1 p Xt(Yt Xt) 3 7 5 .

Proof. The proof of this proposition comes from the computation of the dynamics of the generic compo-nent of the process (Yn,Xn).

dXtrYts= rXtr 1YtsdXt+ sXtrYts 1dYt+ 1 2r(r 1)X r 2 t Ytsd < X >t = (rbX_tr 1Y_ts+1+ r Xr tYts)dt + rX r 1 t Yts p Xt(Yt Xt)dWt s Yts 1X r+1 t dt+ + 2_r(r ₁₎ 2 Y s tXtr 1(Yt Xt)dt = ✓ rbXtr 1Yts+1+ r XtrYts s Yts 1Xtr+1+ 2_r(r ₁₎ 2 Y s+1 t Xtr 1 2_r(r ₁₎ 2 Y s tXtr ◆ dt+ + rXtr 1Yts p Xt(Yt Xt)dWt.

The dynamics of Yn is found by setting r = 0 and s = 1, . . . , n, the dynamics of Xn is determined by

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The approximated price of a guaranteed annuity option will be GAOx(t, T )m= g ⇣tE ⇥ ⇣Tfm(YT, XT)Ft⇤ = g ⇣tE ✓ ↵(T ) + !(T )  YT XT ◆ fm(YT, XT)Ft = g ⇣tE ⇥ a0g(T )Hn+1(YT, XT)Ft⇤ = g ⇣t a0g(T )exp ˜An+1(T t)  Yt,n+1 Xt,n+1 , where a0

g(T )is obtained following equation (2.29).

2.4.3 LHCC(2)

Let us now consider the linear hypercube cascade model, introduced in [1], with m = 2. The dynamics of the process (Y, X) is

dYt= 11Xtdt dX1t= 1(✓1X2t X1t)dt + 1 p X1t(Yt X1t)dW1t dX2t= 2(✓2Yt X2t)dt + 2 p X2t(Yt X2t)dW2t d <W1, W2>t= 0,

for some 1 0and , ✓, 2 R2+ such that

✓i 1 1

i

f or i = 1, 2. The matrices and b are respectively

=  1 1✓1 0 2 , b =  0 2✓2 .

In this way conditions stated in Proposition 2.29 are satisfied (see [1]), in other words bi X ji j = 1{i=m}m✓m= 1_{{i=m} mm} 0, i+ ii+ bi+ X ji ( j+ ij)+= 1 i+ i✓i = 1+ ii+ 1i6=m i,i+1+ 1i=mbm 0.

The advantage of the cascade model is that it allows default intensity values to persistently be close to zero over extended periods of time. It also allows to work with a multidimensional model parsimoniously as the number of free parameters is equal to 3m + 1 (6 in our case) whereas it is equal to 3m + m2 _for

the standard LHC model.

Survival and Death Benefit

As the previous section, we can find a formula for the survival and death benefit using the cascade model. The first thing to do is to define our process (X2,Y2)

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Proposition 2.34. The dynamics of (Y2,X2) is dYt,2= ˜0Xt,2dt (2.37) d_Xt,2= (˜bYt,2+ ˜Xt,2)dt + ˜⌃(Yt,2,Xt,2)dWt, (2.38) where ˜0 =  1 1 0 0 0 0 0 0 0 0 2 1 0 2 1 0 , ˜b = 2 6 6 6 6 6 6 6 6 4 0 0 2✓2 0 0 0 0 0 0 0 0 2✓2 0 0 3 7 7 7 7 7 7 7 7 5 , ˜ = 2 6 6 6 6 6 6 6 6 4 1 1✓1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 21 12 21 0 0 21✓1 0 0 1 1 0 1✓1 1 0 0 0 0 22 22 22✓2+ 22 0 0 0 0 0 1 2 1 0 0 0 2✓2 1✓1 0 1 2 3 7 7 7 7 7 7 7 7 5 , ˜ ⌃(Yt,2,Xt,2) = 2 6 6 6 6 6 6 6 6 6 4 1 p X1t(Yt X1t) 0 0 2 p X2t(Yt X2t) 1X1tpX1t(Yt X1t) 0 1Yt p X1t(Yt X1t) 0 0 2X2t p X2t(Yt X2t) 0 2YtpX2t(Yt X2t) 1X2t p X1t(Yt X1t) 2X1t p X2t(Yt X2t) 3 7 7 7 7 7 7 7 7 7 5 , dWt=  dW1t dW2t .

Proof. The proof is the same as Proposition 2.32.

We can now compute the price of the survival benefit at time t  T , SBt(T ) = C1{⌧>t} ⇣t E ⇥ ↵(T )YT+ !1(T )YT2+ !2(T )XTYT Ft⇤ = C1{⌧>t} ⇣t E ⇥ ↵(T )YT+ !1(T )YT2+ !21(T )X1TYT+ !22(T )X2TYT Ft⇤ = C1{⌧>t} ⇣t a0s(T )exp ˜A2(T t)  Yt,2 Xt,2 , where a0 s(T ) = [↵(T ) !1(T ) 0 0 0 !21(T ) 0 !22(T ) 0]and ˜A2= ₀ _˜ ˜b ˜ . The t-price of the death benefit is

DBt(T ) = C1{⌧>t} ⇣t Z T t E ⇥

↵(u)X1u+ ↵(u)X2u+ !1(u)X1uYu+ !1(u)X2uYu+ !21(u)X1u2 + !22(u)X2u2 Ft⇤du

= C1{⌧>t} ⇣t Z T t a0d(u)exp A2(u t)  Yt,2 Xt,2 du, where a0

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Guaranteed annuity option

For the GAO we can follow the same procedure as before, so we will study the process (Yn,Xn), where

Yn=  Y Y2 . . . Yn Xn=  X1 X2 X12X1Y . . . X13 X12Y X1Y2 X23X22Y X2Y2X1X2Y . . . X1X2Yn 2 .

Proposition 2.35. The dynamics of (Yn,Xn)is

dYt,n = ˜0Xt,ndt (2.39) d_Xt,n = (˜bYt,2+ ˜Xt,2)dt + ˜⌃(Yt,n,Xt,n)dWt. (2.40) where ˜ 2 RdY⇥dX , ˜b 2 RdX⇥dY , ˜ 2 RdX⇥dX and ˜⌃ 2 RdX⇥2_. ˜0 = 2 6 6 6 4 1 1 0 0 0 0 . . . 0 . . . 0 . . . 0 0 0 2 1 0 2 1 . . . 0 . . . 0 . . . ... ... ... ... ... ... ... ... ... ... ... 0 0 0 0 0 0 . . . n 1 . . . n 1 . . . 3 7 7 7 5, ˜b = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 . . . 0 . . . 0 2✓2 0 . . . 0 . . . 0 0 0 . . . 0 . . . 0 0 0 . . . 0 . . . 0 0 0 . . . 0 . . . 0 ... ... ... ... ... ... 0 0 . . . 2✓2 . . . 0 ... ... ... ... ... ... 3 7 7 7 7 7 7 7 7 7 7 7 7 5 , ˜ = 2 6 6 6 4 1 1✓1 0 . . . 0 . . . 0 . . . 0 0 0 0 2 0 . . . 0 . . . 0 . . . 0 0 0 . . . ... ... ... ... ... . . ... 0 0 0 2✓2 . . . 1✓1 . . . (n 2) 1 (n 2) 1 1 2 3 7 7 7 5, ˜ ⌃(Yt,2,Xt,2) = 2 6 6 6 4 1pX1t(Yt X1t) 0 0 2 p X2t(Yt X2t) ... ... 1X2tYtn 2 p X1t(Yt X1t) 2X1tYtn 2 p X2t(Yt X2t) 3 7 7 7 5, dW_t=  dW1t dW2t .

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process, so we will take Xi 1X

j

2Yr with i, j and r such that 0 < i + j + r  n,

d(X1ti X j 2tYtr) = iX1ti 1X j 2tYtrdX1t+ jX1ti X j 1 2t YtrdX2t+ rX1ti X j 2tYtr 1dYt+ +i(i 1) 2 X i 2 1t X j 2tYtrd < X1>t+j(j 1) 2 X i 1tX j 1 2t Ytrd < X2>t = i1(✓1X1ti 1X j+1 2t Ytr X1ti X j 2tYtr)dt + i 1X1ti 1X j 2tYtr p X1t(Yt X1t)dW1t+ + j2(✓2X1ti X j 1 2t Ytr+1 X1ti X j 2tYtr)dt + j 2X1ti X j 1 2t Ytr p X2t(Yt X2t)dW2t+ r 1(X1ti+1X j 2tY r 1 t + X1ti X2tj+1Y r 1 t )dt + 2 1i(i 1) 2 (X i 1 1t X j 2tY r+1 t + X1ti X2tjYtr)dt+ + 2 2j(j 1) 2 (X i 1tX j 1 2t Ytr+1+ X1ti X j 2tYtr)dt =  i1(✓1X1ti 1X j+1 2t Ytr X1ti X j 2tYtr) + j2(✓2X1ti X j 1 2t Ytr+1 X1ti X j 2tYtr)+ r 1(X1ti+1X j 2tY r 1 t + X1ti X j+1 2t Y r 1 t ) + 2 1i(i 1) 2 (X i 1 1t X j 2tY r+1 t + X1ti X j 2tY r t)+ + 2 2j(j 1) 2 (X i 1tX j 1 2t Ytr+1+ X1ti X j 2tYtr) dt + i 1X1ti 1X j 2tYtr p X1t(Yt X1t)dW1t+ + j 2X1ti X j 1 2t Ytr p X2t(Yt X2t)dW2t.

The dynamics of Yn can be found by setting i = j = 0 and r = 1, ..., n and the one of Xn can be

computed by choosing i, j such that i + j > 0.

The approximated price of a guaranteed annuity option will be GAOx(t, T )m= g ⇣tE ⇥ ⇣Tfm(YT, XT)Ft⇤ = g ⇣tE ✓ ↵(T ) + !(T )  YT XT ◆ fm(YT, XT)Ft = g ⇣tE ⇥ a0gHn+1(YT, XT)Ft⇤ = g ⇣t a0g(T ) exp ˜An(T t)  Yt,n+1 Xt,n+1 , where a0

g(T )is again obtained following equation (2.29).

2.4.4 Numerical Example

We will present here some numerical experiments concerning the pricing of the aforementioned insurance products in the LHC(1) framework. We will proceed as follows:

i) we will set our model

ii) we will perform an estimation of the prices of the survival, death benefit and the GAO, using the closed formula found in Section 2.4.2

iii) we will carry out a Monte Carlo simulation, in order to show that the price of the GAO found in (2.32) approximates well its real price given by (2.27).

For the specification of the state price density we will follow [19], ⇣t= exp( ↵t) ✓ + 0  Yt Xt ◆

where ↵, and are real valued parameters chosen such that ⇣t> 0for every t > 0.

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can be estimated by calibration, which consists in finding the values of the parameters which minimise a mean squared error function, or alternatively by performing a quasi-maximum likelihood estimation or a generalised method of moments estimation.

The problem of the estimation of the parameters is outside the scope of this work, this is they are obtained by calibrating the model to the inverse of benchmark portfolio (the MSCI world index) and the longevity index (LLMA index related to the German population) with a sample from January 1970 to January 2013. For the value of the benefit C and the annual payment g (involved in the GAO price), we set with-out loss of generality C = 100 and g = 1.

For the approximation of the payoﬀ concerning the GAO, as it involves, under the assumptions of the LHC(1), the positive part of a polynomial of order 2, we decided to use a polynomial of the same order.

In order to do that we will proceed by using a second order Taylor series around the points x = 0 and y = 0.

We recall that for a two variables function f(x, y), its Taylor series to second order around the point (a, b)is equal to

f (x, y) = f (a, b) + fx(a, b)(x a) + fy(a, b)(y b) +

1

2fxx(a, b)(x a)

2₊1

2fyy(y b)

2₊

+ fxy(a, b)(x a)(y b) + o((x a)(y b))

where f·is the first order derivative of f, f·· is the second order derivative of f and o((x a)(y b))is

the Peano form of the remainder.

The approximation of the GAO payoﬀ is performed with MatLab R2016b, where we used the built-in function taylor(f, order, point), which takes three input arguments: the function to approximate, the order of approximation and the expansion point. This procedure gives us the following result (for a whole life annuity of 10 years)

✓ ax(T ) 1 g ◆+ ⇡ 5.17 + 0.98XT + 3.14YT + 1.23XT2+ 2.67YT2+ 0.6XTYT.

Remark 2.36. Even if the (·)+_{function is not diﬀerentiable everywhere, this does not cause any problem}

from a numerical point of view.

Table (2.11) provides the prices of the survival, the death benefit and the GAO for an individual of age x at time t = 0 and for diﬀerent maturities T = {0, 10, 20, 30, 40, 50, 60, 70}. Figure (2.1), Figure (2.2) and Figure (2.3) show us the behaviour of this life insurance product prices with respect to the maturity T . Maturity T SB0(T ) DB0(T ) GAOx(0, T )m 0 100 0 7.8494 1 91.6605 4.9101 5.8915 10 61.4419 36.4830 3.7564 20 37.2664 61.7419 2.2785 30 22.6032 77.0622 1.3820 40 13.7095 86.3544 0.8383 50 8.3153 91.9904 0.5084 60 5.3020 95.4089 0.3084 70 3.0590 97.4822 0.1870

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Figure 2.1: SB price as a function of the maturity

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Figure 2.3: GAO price as a function of the maturity

We can split our comments in three parts, one for each insurance product we evaluated

i) concerning the survival benefit we can notice, observing the second column of table (2.1) and figure (2.1), that it is a decreasing function of the maturity, which is quite trivial, because as time goes by, the probability of surviving decreases, so does the price.

ii) the same reasoning, but reversed, can be done about the death benefit. Looking at the third column of the table (2.1) and figure (2.2), we notice that it is an increasing function of the time T . As the maturity rolls away from the initial date t = 0, the probability of surviving decreases, which has as a consequence the fact that the price of this benefit increases over time.

iii) finally, observing the last column of table (2.1) and figure (2.3), we can notice that the price of the guaranteed annuity option decreases with respect to the term of this contract. This can be explained by the fact that this option contains the sum of survival benefits, so, as the survival benefit is a decreasing function of T , the GAO as well will have the same behaviour.

Even if these comments are quite trivial, this means that the model we built, prices correctly our insurance products, at least the survival and death benefit, as we used their exact closed formula for their pricing. Monte Carlo Simulation We can now carry out a Monte Carlo simulation, in order to check if the approximated formula given by (2.30), gives us a good approximation of the real price of the GAO. To do that we have to deal with the discretization of our stochastic process (Y, X), given by (2.31) (2.32). In an attempt to achieve this goal, we have to be careful with the diﬀusion part of X, which involves a square root, which has to contain positive values only. In order to overcome this problem, there are several possibilities, also presented in [33], which consist in forcing the value of the discretized process to be positive. One choice consists in taking the absolute value of the process who contains the square root, the process X, then in this case, the Euler scheme would be written as

Yt+h= Yt hXt

Xt+h=|Xt+ (bYt+ Xt)h +

p

Xt(Yt Xt)B|

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The convergence of this scheme is assured by [4] and [7] who proved the weak and strong convergence of such a scheme in a general setting.

We choose a time step h = 0.1.

Table (2.2) shows the results of our study for a number of simulation N = {1000, 10000} and the absolute error computed with respect the results provided in table 2.1.

Maturity T N=1000 ✏1000 CI 95% N=10000 ✏10000 CI 95% 0 8.1462 0.2968 (7.7683, 8.5241) 8.1459 0.2965 (7.7789, 8.5129) 1 6.1284 0.2369 (5.6631, 6.5935) 6.1282 0.2367 (5.6775, 6.5789) 10 3.9075 0.1515 (3.7321, 4.0829) 3.9075 0.1515 (3.7400, 4.0750) 20 2.3701 0.0916 (2.2578, 2.483) 2.3700 0.0915 (2.2580, 2.482) 30 1.4375 0.0555 (1.2987, 1.5763) 1.4375 0.0555 (1.2989, 1.5761) 40 0.8719 0.0337 (0.6506, 1.0932) 0.8719 0.0337 (0.6566, 1.0872) 50 0.5288 0.0204 (0.3309, 0.7267) 0.5288 0.0204 (0.3334, 0.7242) 60 0.3207 0.0123 (0.1804, 0.4610) 0.3205 0.0121 (0.1819, 0.4591) 70 0.1945 0.0075 (0.0724, 0.3166) 0.1943 0.0073 (0.0771, 0.3115) Table 2.2: Monte Carlo simulation with confidence intervals at 95% and absolute error of the guaranteed annuity option price

The first thing we can notice, comparing the last column of table (2.1) and table (2.2), is that the approximated formula for the price of the guaranteed annuity option underestimates the price found with the Monte Carlo method, and this for every maturity T .

We can also see that ✏, which represents the absolute error between the approximated and the Monte Carlo prices is consistent and it is a decreasing function of the maturity T . In addition the value of ✏ seems to be independent from the number of sumulations N.

This means that our approximation does not provide a good estimation of the price of a GAO, this can be due to the fact that maybe we are using a polynomial with a low degree (the order two in this experiment). So, we can try to see what happens if we approximate the payoﬀ of the GAO with a higher degree polynomial. According to [1], the price approximation stabilises rapidly such that a polynomial of degree 10 appears to be accurate in order to provide a good approximation of our payoﬀ.

In this case our approximated function, always found using a Taylor approximation, will contain 66 terms, as we deal with a ten degree polynomial of two real positive variables x and y.

Table (2.3) presents, for diﬀerent maturities T , the approximated price of the GAO option when using a ten degree polynomial.

Maturity T GAOx(t, T )m ✏1000 ✏10000 0 8.1453 0.0009 0.0006 1 6.1287 0.0003 0.0005 10 3.9067 0.0008 0.0008 20 2.3696 0.0005 0.0004 30 1.4381 0.0006 0.0006 40 0.8726 0.0007 0.0007 50 0.5283 0.0005 0.0005 60 0.3198 0.0009 0.0009 70 0.1949 0.0004 0.0004

Table 2.3: Approximated price of a GAO using a ten degree polynomial with the absolute error Remark 2.37. The value of ✏ has been computed with respect to the Monte Carlo prices found in table (2.2).

As we can see from this table, the absolute error seems to be significantly lower than before, which means that this approximation works better than the previous one. In addition, we can remark that the error does not seem to be dependent on the maturity of the insurance contract, as we remarked in the last experiment with the two order polynomial.

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Sensitive Analysis In this paragraph we will carry out a sensitive analysis by perturbing, one at a time, the parameters that aﬀect the survival probability involved in the approximated GAO price: ( , b, ). We focused on the price of a guaranteed annuity option with maturity T = 10.

Figure (2.4), (2.5) and (2.6) shows the sensitivity of GAOx(0, 10) with respect to the parameters , b

and .

Figure 2.4: GAO price as a function of

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Figure 2.6: GAO price as a function of

In Figure (2.4) we show the impact of the parameters on the price of the GAO. We can notice that as increases, the value of the insurance contracts decreases, but very slowly. This is reasonable, as we can see in Remark 2.28, the value of the mortality intensity is an increasing function of , so the bigger µtis, the smaller will be the survival probability.

The range of the parameter has be chosen in order to satisfy the conditions stated in Remark 2.31, as the other parameters (b and ) are fixed.

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Conclusion

We worked in a linear rational framework in order to price a family of insurance liabilities. To achieve this goal, we focus on the pricing of the quantities which are building blocks for these products, as well other class of life insurance conracts which are functions of these building blocks.

One of the main features of this framework is that under the hypothesis that the benefit is a constant value (which, despite its simplicity, covers a set of contracts that are actually traded on the market) or a polynomial function of a linear drift process, we are always able to calculate the exact price of the survival and death benefit, thanks to the result found in Theorem 2.14. This is also possible in a linear rational framework where the coefficients of the mortality intensity are not anymore constant, but time dependent. This was motivated by the fact that in practice our model should be able to replicate the observed trend in mortality, which cannot always be done if we consider models with constant coefficients. The most important feature of this work is that, even when considering an insurance product whose payoff is not necessarily a linear function, we are capable, under the condition that our state space is compact, to approximate this function by a polynomial, which allows us to benefit once again of the result of Theorem 2.14.

This is very important because, instead of treating a complex function, which can involve a very large number of terms (see for instance the whole life annuity (2.25)), we treat a polynomial, which is easier to handle in our setting.

This advantage of the model is then illustrated in a numerical example at the end of this work. We showed that the choice of the degree of the polynomial which approximates the payoﬀ function is a crucial step when pricing a GAO: a low degree, such as two, cannot guarantee a good approximation, if compared to the Monte Carlo price, its absolute error being a decreasing function of the maturity of the contract and it does not depend on the number of simulations we have performed.

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Part II

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Introduction

BPI France Financement

The Banque publique d’investissement or Bpifrance is a french institution who finances the enterprises. Its aim is to support the french firms, specially the small and the mid size enterprises, but also the big ones, when they have a dimension such that they are important for the national economy or the employment.

Bpifrance supports the french enterprises from the beginning to the quotation in Stox exchange, proposing diﬀerent solutions:

i) co-financing the other banks to support diﬀerent projects.

ii) funding of innovation: Bpifrance supports innovating processes through financial aids, or in the form of equity participation.

iii) seizure of new markets in France or in foreign countries: Bpifrance supports export projects (de-velopment, implantation) with UBIFRANCE and Coface, partners for Bpifrance Export.

iv) investments in own funds: Bpifrance invests, alongside public and private actors, in investement capital funds, which invest in SMEs.

v) collateral for the other banks: Bpifrance provides a guarantee according to the financial projects to encourage the SMEs during the more risky phases.

Guarantee Activity

Bpifrance provides banks with a guarantee from 40% to 70% of the loan value, depending on the financial projects, to encourage them to finance SMEs, in the most risky phases. The banks, benefiting from the guarantee, pay a guarantee premium (commission) which is calculated according to the purpose of the project and the amount of financing. The commission can be collected at the starting date of the loan or collected, over time. In case of default, the payment is interrupted.

Several guarantee funds are backed by the diﬀerent coverages oﬀered by the Bpifrance guarantee. They are classified in National Funds (Creation, Transmission, Development, for example) and Regional. Several flows debit or credit a guarantee fund. Indeed, a guarantee fund is credited by:

i) The fees paid by the banks for the Bpifrance guarantee (it only covers part of the Bpifrance risks). ii) The Government’s endowment to cover losses with a probability of 70%.

iii) The financial products that are resources resulting from the management of the assets of the guarantee fund.

Conversely, a guarantee fund is also debited by:

i) Possible compensation when the guarantee is involved.

LINEAR RATIONAL INSURANCE MODEL & ECONOMIC SCENARIO GENERATOR

SCUOLA DI DOTTORATO

Dipartimento di / Department of

STATISTICA E METODI QUANTITATIVI

Dottorato di Ricerca in / PhD program STATISTICA E FINANZA MATEMATICA

Ciclo / Cycle XXXII

Curriculum in

MATEMATICA PER LA FINANZA

LINEAR RATIONAL INSURANCE MODEL

&

ECONOMIC SCENARIO GENERATOR

Cognome / Surname

DEL GUSTO Nome / Name LUIGI CARMINE

Matricola / Registration number

746006

Tutore / Tutor: EMANUELA ROSAZZA GIANIN

Coordinatore / Coordinator: GIORGIO VITTADINI

Ph.D. thesis

Subject: LINEAR RATIONAL INSURANCE MODEL

ECONOMIC SCENARIO GENERATOR

Del Gusto

Luigi Carmine

Contents

I Linear Rational Insurance Model

5

II Economic Scenario Generator

39

Part I

Introduction

Chapter 1

Essential notions

1.1 State Price Density

1.2 Linear Rational Term Structure Models

1.3 Polynomial Diﬀusion Models

Chapter 2

Linear Rational Insurance Model

2.1 Financial Market

2.2 Mortality Model

2.2.1 Modelling of the Mortality Force

2.3 Valuation of insurance contracts

2.3.1 Survival benefit

2.3.2 Death benefit

2.3.3 Whole Life Annuity

2.3.4 Guaranteed annuity option

2.4 Application

2.4.1 The Linear Hypercube Model

2.4.2 LHC(1)

2.4.3 LHCC(2)

2.4.4 Numerical Example

Conclusion

Part II

Introduction

BPI France Financement

Guarantee Activity